Preview — Love and Math
by Edward Frenkel

“People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“The laws of Nature are written in the language of mathematics.” Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“The interaction between math and physics is a two-way process, with each of the two subjects drawing from and inspiring the other. At different times, one of them may take the lead in developing a particular idea, only to yield to the other subject as focus shifts. But altogether, the two interact in a virtuous circle of mutual influence.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the “initiated few.” It belongs to all of us.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“The fact that these highly abstract notions coalesce in such refined harmony is absolutely mind-boggling. It points to something rich and mysterious lurking beneath the surface, as if the curtain had been lifted and we caught glimpses of the reality that had been carefully hidden from us. These are the wonders of modern math, and of the modern world.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“I will also talk about my experience of growing up in the former Soviet Union, where mathematics became an outpost of freedom in the face of an oppressive regime. I was denied entrance to Moscow State University because of the discriminatory policies of the Soviet Union. The doors were slammed shut in front of me. I was an outcast. But I didn’t give up. I would sneak into the University to attend lectures and seminars. I would read math books on my own, sometimes late at night. And in the end, I was able to hack the system. They didn’t let me in through the front door; I flew in through a window. When you are in love, who can stop you?”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“As the story goes, Albert Einstein’s wife Elsa remarked, upon hearing that a telescope at the Mount Wilson Observatory was needed to determine the shape of space-time: “Oh, my husband does this on the back of an old envelope.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It's as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“How to describe the excitement I felt when I saw this beautiful work and realized its potential? I guess it's like when, after a long journey, suddenly a mountain peak comes in full view. You catch your breath, take in its majestic beauty, and all you can say is "Wow!" It's the moment of revelation. You have not yet reached the summit, you don't even know yet what obstacles lie ahead, but its allure is irresistible, and you already imagine yourself at the top. It's yours to conquer now. But do you have the strength and stamina to do it?”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“The fact that such objective and enduring knowledge exists (and moreover, belongs to all of us) is nothing short of a miracle. It suggests that mathematical concepts exist in a world separate from the physical and mental worlds”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“a drunkard may not know which number is larger, 2/3 or 3/5, but he knows that 2 bottles of vodka for 3 people is better than 3 bottles of vodka for 5 people.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“As someone told me later, writing papers was the punishment we had to endure for the thrill of discovering new mathematics. This was the first time I was so punished.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“It’s rare, he says, that we “encounter a person who asserts vehemently that the mere thought of reading a novel, or looking at a picture, or seeing a movie causes him insufferable torment,” but “sensible, educated people” often say “with a remarkable blend of defiance and pride” that math is “pure torture” or a “nightmare” that “turns them off.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“It turned out that Evgeny Evgenievich indeed had a clever plan how to convert me to math. As soon as I came to his office, he asked me, “So, I hear you like quantum physics. Have you heard about Gell-Mann’s eightfold way and the quark model?” “Yes, I’ve read about this in several popular books.” “But do you know what was the basis for this model? How did he come up with these ideas?” “Well...” “Have you heard about the group SU(3)?” “SU what?” “How can you possibly understand the quark model if you don’t know what the group SU(3) is?”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“If you experience this feeling once, you will want to go back and do it again. This was the first time it happened to me, and like the first kiss, it was very special. I knew then that I could call myself mathematician.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“In 1940, during the war, André Weil was imprisoned in France for refusing to serve in the army. As the obituary published in The Economist put it,1 [Weil] had been deeply struck.... by the damage wreaked upon mathematics in France by the first world war, when “a misguided notion of equality in the face of sacrifice” led to the slaughter of the country’s young scientific elite. In the light of this, he believed he had a duty, not just to himself but also to civilization, to devote his life to mathematics. Indeed, he argued, to let himself be diverted from the subject would be a sin. When others raised the objection “but if everybody were to behave like you...”, he replied that this possibility seemed to him so implausible that he did not feel obliged to take it into account.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“Perhaps, a bigger point is that it is perfectly OK if something is unclear. That’s how I feel 90 percent of the time when I do mathematics, so welcome to my world! The feeling of confusion (even frustration, sometimes) is an essential part of being a mathematician. But look at the bright side: how boring would life be if everything in it could be understood with little effort! What makes doing mathematics so exciting is our desire to overcome this confusion; to understand; to lift the veil on the unknown. And the feeling of personal triumph when we do understand something makes it all worthwhile.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“Braid groups have many important practical applications. For example, they are used to construct efficient and robust public key encryption algorithms.7 Another promising direction is designing quantum computers based on creating complex braids of quantum particles known as anyons. Their trajectories weave around each other, and their overlaps are used to build “logic gates” of the quantum computer.8 There are also applications in biology. Given a braid with n threads, we can number the nails on the two plates from 1 to n from left to right. Then, connect the ends of the threads attached to the nails with the same number on the two plates. This will create what mathematicians call a “link”: a union of loops weaving around each other. In the example shown on this picture, there is only one loop. Mathematicians’ name for it is “knot.” In general, there will be several closed threads. The mathematical theory of links and knots is used in biology: for example, to study bindings of DNA and enzymes.9 We view a DNA molecule as one thread, and the enzyme molecule as another thread. It turns out that when they bind together, highly non-trivial knotting between them may occur, which may alter the DNA. The way they entangle is therefore of great importance. It turns out that the mathematical study of the resulting links sheds new light on the mechanisms of recombination of DNA. In mathematics, braids are also important because of their geometric interpretation. To explain it, consider all possible collections of n points on the plane. We will assume that the points are distinct; that is, for any two points, their positions on the plane must be different. Let’s choose one such collection; namely, n points arranged on a straight line, with the same distance between neighboring points. Think of each point as a little bug. As we turn on the music, these bugs come alive and start moving on the plane. If we view the time as the vertical direction, then the trajectory of each bug will look like a thread. If the positions of the bugs on the plane are distinct at all times – that is, if we assume that the bugs don’t collide – then these threads will never intersect. While the music is playing, they can move around each other, just like the threads of a braid. However, we demand that when we stop the music after a fixed period of time, the bugs must align on a straight line in the same way as at the beginning, but each bug is allowed to end up in a position initially occupied by another bug. Then their collective path will look like a braid with n threads. Thus, braids with n threads may be viewed as paths in the space of collections of n distinct points on the plane.10”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“had been deeply struck.... by the damage wreaked upon mathematics in France by the first world war, when “a misguided notion of equality in the face of sacrifice” led to the slaughter of the country’s young scientific elite. In the light of this, he believed he had a duty, not just to himself but also to civilization, to devote his life to mathematics. Indeed, he argued, to let himself be diverted from the subject would be a sin. When others raised the objection “but if everybody were to behave like you...”, he replied that this possibility seemed to him so implausible that he did not feel obliged to take it into account.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“When I first started coming to the seminar, Gelfand had a young physicist, Vladimir Kazakov, present a series of talks about his work on so-called matrix models. Kazakov used methods of quantum physics in a novel way to obtain deep mathematical results that mathematicians could not obtain by more conventional methods. Gelfand had always been interested in quantum physics, and this topic had traditionally played a big role at his seminar. He was particularly impressed with Kazakov’s work and was actively promoting it among mathematicians. Like many of his foresights, this proved to be golden: a few years later this work became famous and fashionable, and it led to many important advances in both physics and math. In his lectures at the seminar, Kazakov was making an admirable effort to explain his ideas to mathematicians. Gelfand was more deferential to him than usual, allowing him to speak without interruptions longer than other speakers. While these lectures were going on, a new paper arrived, by John Harer and Don Zagier, in which they gave a beautiful solution to a very difficult combinatorial problem.6 Zagier has a reputation for solving seemingly intractable problems; he is also very quick. The word was that the solution of this problem took him six months, and he was very proud of that. At the next seminar, as Kazakov was continuing his presentation, Gelfand asked him to solve the Harer–Zagier problem using his work on the matrix models. Gelfand had sensed that Kazakov’s methods could be useful for solving this kind of problem, and he was right. Kazakov was unaware of the Harer–Zagier paper, and this was the first time he heard this question. Standing at the blackboard, he thought about it for a couple of minutes and immediately wrote down the Lagrangian of a quantum field theory that would lead to the answer using his methods. Everyone in the audience was stunned.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“In fact, it is virtually impossible for students to do their own research without someone guiding their work. Having an advisor is absolutely essential.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality

“In his thoughtful essay about Taniyama, Shimura made this striking comment:18 Though he was by no means a sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes.”
―
Edward Frenkel,
Love and Math: The Heart of Hidden Reality